Class 09/13/2021

Example of First Order Linear System

Rewrite the initial value problem

$$\begin{gathered} \frac{d^{3}y}{d{x}^3}-x^2\frac{d^{2}y}{d{x}^2}+3x\frac{dy}{dx}+5y=e^{-4x} \\ y(0)=1,\frac{dy}{dx}(0)=3,\frac{d^{2}y}{d{x}^2}(0)=7 \end{gathered}$$

Rewrite as a first order linear system

Solution

1. Make the change of variables

   1. $y_1=y,y_2=\frac{dy}{dx},y_3=\frac{d^{2}y}{d{x}^2}$

2. Rewrite initial equation

   1. $\frac{d^{3}y}{d{x}^3}=e^{-4x}-5y_1-3x{y}_2+x^2{y}_3$

3. Put into matrix form. The matrix is composed of the factors multiplied to the $y1,y2,y3$ terms.

4. $$\begin{gathered} \frac{d}{dx}\left[\begin{array}{c}y1\\ y2\\ y3\end{array}\right]=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 1\\ 5& -3x& x^2\end{array}\right]\left[\begin{array}{c}y1\\ y2\\ y3\end{array}\right]+\left[\begin{array}{c}0\\ 0\\ e^{-4x}\end{array}\right] \\ \left[\begin{array}{c}y1(0)\\ y2(0)\\ y3(0)\end{array}\right]=\left[\begin{array}{c}1\\ 3\\ 7\end{array}\right] \end{gathered}$$

Introduction to First Order Nonlinear Systems

Consider the IVP (system of first order nonlinear differential equations):

This can also be written as

System would be linear if $\vec{f}(x,\vec{y})=P(x)\vec{y}+\vec{g}(x)$

Autonomous if the right hand side is independent of x. $\frac{d}{dx}\vec{y}=\vec{f}(\vec{y})=\left[\begin{array}{c}f1(x,y1,y2,yn)\\ f2(x,y1,y2,yn)\\ fn(x,y1,y2,yn)\end{array}\right]$

Existence Uniqueness Theorem (EUT) for nonlinear systems

In the $(n+1)$ - dimensional open rectangular $(x,y1,yn)ϵR=a<x<b1,\alpha 1$

Example

Find the largest rectangular region we know there exists a unique solution to the IVP

This IVP is a third order nonlinear differential equation. First we rewrite as a first order system with the change of variables.

$y1=y,y2=\frac{dy}{dx},y3=\frac{d^{2}y}{d{x}^2}$

We combine the continuity of $\vec{f}(x,\vec{y})$ and $J(x,\vec{y})$